Quasi - Variational Inequality Formulations and Solution Approaches for Dynamic User Equilibria

1. Quasi - Variational Inequality Formulations and Solution Approaches for Dynamic User Equilibria Xuegang (Jeff) Ban Civil & Environmental Engineering Department University of Wisconsin – Madison ECE 7750 Utah State University January 24, 2005

2. Outline • Introduction and Background • Optimal Control Formulation for DUE • Path-based Discrete-time DUE Model • Link-based Discrete-time QVI Model • Decomposition Schemes for Link-based Model • Numerical Examples on I-394 Corridor Network • Conclusions and Future Study

3. Introduction and Background

4. Transportation Systems • Components • Road Network • Traffic Control Devices (Signs, Signals,…) • Vehicles • People • Environment (Weather,…) • Characteristics • Nonlinear: travel cost is highly nonlinear in terms of traffic flow • Dynamic: OD demand, travel time,… • Uncertain: network, behavior, perception Supply OD trip demands, behavior

5. Signal Control Systems Electronic Road Pricing Intelligent Parking Traffic Monitoring and Operation Center Travel Time Prediction Ramp Metering Intelligent Transportation Systems (ITS) ITS integrates advanced information technologies, communication, and computer and electronical techniques into vehicles, road networks, and transportation management centers, with the purpose to alleviate traffic congestion, improve safety and capacity, reduce vehicle emissions, and finally enhance the productivity of the entire transportation system. Intelligent Vehicle ATIS Incident Detection

6. DTA and ITS • Dynamic Traffic Assignment (DTA) Capability of estimation of current and prediction of future traffic states National ITS Architecture, from FHWA DTA Project Web Page DTA

7. Dynamic User Equilibrium (DUE) • A Special and Most Important Problem of DTA DUE is to predict the dynamic traffic patterns, i.e., the time-varying link flows, link inflow rate and exit flow rate, and then travel time and speed for each link, given the network and time-dependent OD flows, by assuming that each of the motorists is trying to minimize his/her individual travel time.

8. Optimal Control Formulation for DUE

9. Dynamic Network Constraints : inflow rate into link a for flows between OD pair rs along path p at time t : exit flow rate from link a… : total number of vehicles (link flows) on link a… : link travel time for flows entering link a at time t.

10. Dynamic Network Constraints (Con’d) • Definitional Constraints: • Non-negativity Condition: • Boundary Condition:

11. Link and Path Travel Time Function • Link Performance (Travel Time) Function Whole-link model (the simplest form) • Path Travel Time (actual experienced travel time) Not close-formed; has to be calculated recursively.

12. Optimality Condition • DUE Condition (Ran and Boyce, 1996): If, for each OD pair at each instant of time, the actual travel times experienced by travelers departing at the same time are equal and minimal, the dynamic traffic flow over the network is in a travel-time-based dynamic user equilibrium (DEU) state.

13. Optimal Control Model for DUE • Find such that subject to • Link Travel Time: • Path Travel Time:

14. Current Research for DUE • Literature Review of DUE Models • MP: Merchant and Nemhauser (1978a, 1978b) • OC: Friesz (1989) • VI: Friesz (1993), Ran and Boyce (1994, 1996) • QVI: Bliemer and Bovy (2003) • Previous DUE Study • Solution algorithm: discretization • Regular VI formulations were applied which can not capture properly the dynamic nature of DUE, especially the flow propagation • QVI formulation was applied for multi-user-class DUE in a path-based fashion • No feasible algorithm for solving continuous-time DUE directly on practical transportation networks • Can not solve DUE on practical transportation networks with exact solutions.

15. Mathematical Programming Problems • Nonlinear Complementarity Problem (NCP), NCP(F) Given a mapping , find such that • Mixed Complementarity Problem (MiCP), MiCP(G,H) Given G and H being two mappings, and , , find a pair such that • VI (Variational Inequality) Given a subset K of and a mapping , VI(K,F) is to find a vector in such that • QVI (Quasi-Variational Inequality) Find such that where is a point to set mapping. • Merit Function of VIs A merit function for the VI(K, F) is a nonnegative function such that x is a solution to the VI if and only if

16. Research Focus of This Study • Basic DUE Problem Discrete, deterministic, single-user-class DUE problem with fixed demand • Both Path and Link Based QVI Formulations • Decomposition Schemes for Large-Scale DUE Problems • Model Assumptions • Demand: given and fixed (input) • Supply • Network supply: dynamic network constraints, and link and path performance functions • Information Supply: perfect information • Behavior: complementarity condition

17. Path-based QVI Model

18. Path-based Discrete DUE Model (PDDUE) Find Such that • Mass Balance Constraint • Flow Propagation Constraint • Flow Conservation Constraint • Other Constraints s.t.

19. Why a QVI Model? • Flow Propagation Constraints • The defining set is not fixed, but varies with the variables since is itself a function of f • QVI Formulation (Facchinei and Pang, 2003, Page 16) “An extension of a VI in which the defining set of the problem varies with the variable.”

20. PDDUE-QVI Formulations • PDDUE-QVI Find , such that is a solution to the following problem: • is a Point to Set Mapping • Sub-Problem by fixing Find such that • Solution algorithm for PDDUE-QVI An iterative algorithm with an NCP sub-problem being solved at each iteration

21. Relaxed Sub-Problem (RPDDUE) • NCP Sub-Problem • Path Travel Time Function are both fixed matrices Both and are non-decreasing continuously differentiable positive functions with respect to x and u, respectively.

22. Model Properties of RPDDUE • Jacobian Matrix of the Model • Existence Condition RPDDUE has at least one solution • Boundedness Condition The solution set of RPDDUE is nonempty and compact (closed and bounded) if is positive semi-definite. • Local Uniqueness Condition To check if a given solution is locally unique. • Uniqueness Condition No globally unique solution exists for RPDDUE with fixed demand. Not Symmetric or Skew-Symmetric No equivalent NLP Formulation

23. Solution Algorithm for NCP Sub-Problems • PATH solver by Ferris et al. (1995) Major Characteristics • To find a zero of the approximation at each Newton step, an equivalent Linear Complementarity Problem (LCP) is solved. • “Backtracking” approach is used to find the optimal step size. • A non-monotone stabilization scheme and the “watchdog” method were implemented to improve the efficiency and robustness of the solver. • Convergence Results of PATH • Globally convergent solver • Convergence rate: Q-superlinear, and Q-quadratic near the solution • Guarantee Convergence for an NCP with P0 function • Best Available Solver for Medium to Large Scale MiCPs • Efficient and Effective for Solving NCP Sub-Problems

24. Solution Algorithm for PDDUE-QVI • Solution Algorithm • Issues • Searching direction: solution from the relaxed NCP sub-problem • Stopping criterion Error function , where and are the vectors of integer-valued link travel time for each link at each time interval evaluated at and , respectively. • Step size for next iterate Use r an approximate merit function of PDDUE-QVI • Heuristic-based solution algorithm

25. Numerical Examples for PDDUE • Link Travel Time Function Polynomial function in terms of both link inflow rate and total number of vehicles. • OD Demands Function • D3 Network

26. Case Study I • D3 network, K=8 (2 min.) • NCP sub-problems can be solved using PATH efficiently; i.e., less than 1 sec. in a Pentium-III computer with 256 MB memory. • QVI algorithm converges after 3 iterations.

27. Case Study II • Grid network (5 by 5), K = 120 (1 hour) • Number of paths 125: 70; 725: 20 • Initial path flow (Starting point) Assign OD demands evenly to each path. Converges after 2 iterations. Assign all OD demands randomly to one single path. Converges after 6 iterations. Both converge to the same solution. Not sensitive to the starting point. • Results • DUE condition holds, i.e., • Path flows and travel times for P1 and P2 • Link inflow rate, exit flow rate, and total number of vehicles for link 78 P1 P2

28. Scenario II (Conti.) P1 P2 Exit flow Rate (v) Inflow Rate (u)

29. Link-based QVI Model

30. Why Link-based Models? • Pros and Cons of Path-based Models • Pros Straightforward, easy to derive, and simple • Cons Requires the path-numeration, hard to be applied for large-scale problems. e.g., for a 13 by 13 grid network, the max # o paths is nearly 3 million. Column generation? • Link-based Models • Defined on aggregated and disaggregated link inflows • No path-enumeration is needed • Decomposition schemes can be applied for large-scale problems

31. Link-based Discrete DUE (LDDUE) • Link-based DUE Condition If, from each decision node to each destination node at each time interval, the actual travel time for all the routes that are being used are equal and minimal, the dynamic traffic flow over the network is in a travel-time based dynamic user equilibrium (DUE) state. • NCP Expression : the link inflow to link a at time k with respect to the destination s. : the travel time for flows entering link a at time k, a function of u : the minimum travel time from node i to destination s at time k.

32. LDDUE (Cont.) Findsuch that subject to: 1) Mass Balance 2) Flow Conservation 3) Flow Propagation 4) Other Constraints Link Travel Time Function

33. Link-based QVI Model • Link-based QVI Model Find such that where • QVI Formulation (Facchinei and Pang, 2003) Find such that where is a point to set mapping.

34. Model Properties • Model Properties • Existence Condition At least one solution exists for the model • Uniqueness Condition: Multiple solutions may exist due to the fact that the defining set is not fixed • Solution Procedure Iterative solution algorithm with an NCP-formed sub-problem solved at each iteration.

35. NCP Sub-Problem • NCP Formulation where and are fixed matrices; and are smooth functions. • Model and Solution Properties • Solution existence condition At least one solution exists for the sub-problem • Uniqueness condition Unique solution in terms of aggregated flows if is positive definite • Solution Algorithm PATH Solver for single-destination problem

36. Solution Algorithm for Link-based Model • Solution Algorithm • Issues • Stopping criterion Approximate merit function , where and are the vectors of integer-valued link travel time for each link at each time interval evaluated at and , respectively. • Step size for next iterate • Use r an approximate merit function of LDDUE • Newton-based Method

37. Newton-based Method for the Step Size • Newton-based Method (Applied by the PATH Solver) Take the Newton step (step size = 1) as long as the value of the merit function r can be reduced; otherwise, perform a backtracking procedure from the candidate solution to current base inflow to find an optimal step size (with a minimal r value). • Effective to reduce r within several iterations • May be stuck at a local point and then MSA step ( ) is used • Efficient to find an optimal or approximate solution

38. Numerical Examples • Link Performance Function • Testing Networks • Network Specifications D3 network G6 network # of Paths from 1 to 36: 252; from 8 to 36: 70

39. Numerical Examples (Cont.) • Dynamic OD Demands • Case Study I: D3 network with K=120 (30 min.) Converge after 14 iterations and find an optimal solution

40. Numerical Examples (Cont.) • Case Study II: G6 network with K=60 (15 min.) Converge after 16 iterations and find an optimal solution u v

41. Numerical Examples (Cont.) • Case Study III: G6 network with K=120 (30 min.) Solution algorithm terminates with an approximate solution (DUE holds for more than 90% of the times)

42. Decomposition Scheme for Link-based QVI Model

43. Why the Decomposition Schemes? • Problems for Solving Multi-destination DUE • Large dimension • The model is not strictly monotone, and tend to have rank deficiency • Decomposition Schemes for the Relaxed Sub-Problem of LDDUE • Decomposition based on individual destinations • Gauss-Seidel (Sequential) and Jacobi (Parallel) schemes • Similar as the schemes for Static Traffic Assignment (STA)

44. NCP Formulation for Static Traffic Assignment • NCP Formulation for STA with Asymmetric Cost • MiCP Formulation with Side Constraints • NCP Formulation Route Choice Flow Conservation Side Constraint

45. NCP Formulation for STA (Cont.) • Matrix Expression • Assumptions on Link Travel Time t • t is a smooth and strictly monotone positive function in terms of the total link flow . • t is finite as long as is finite. • t is coercive in terms of , i.e., • Properties of t • for a given total link flow vector x. • is strictly monotone if is fixed.

46. NCP Formulation for STA (Cont.) • Properties of the NCP Model • Jacobian matrix of the model is monotone (but not strictly monotone!) • Solution existence condition At least one solution exists for the NCP model • Uniqueness condition Unique solution in terms of the total link flow , but not for the disaggregated link flow .

47. Direct Solving of the NCP Model • PATH Solver on Sioux-Falls Network • Asymmetric Link Cost • Performance • Efficient for small scale problems • Exact solution (1.0e-9)

48. Explicit Proximal Perturbation for Direct Solving • Problem of Direct Solving For Multi-destinations Rank Deficiency • Explicit Proximal Perturbation • Iteratively Solve the Perturbed NCP • Works for Small to Medium Size Problems 2 hours 45 minutes for the Anaheim Network: 416 nodes, 914 links, with 10 destinations (total 38). is a full rank matrix

49. Decomposition Scheme – Gauss-Seidel Method • Decomposition Based on Individual Destinations • Instead of solving a single NCP with size (|A|+|N|)*|S|, solve |S| NCPs with size |A|+|N| which are much easier to solve. • Fix the flows for all other destinations when processing destination s • Decomposed NCP (DNCP) for at Iteration n+1

50. GS Method (Cont.) • Performance of Pure GS • Converges faster than direct solving, and produces the exact solutions • Problems of GS Method • Full step is always taken, may be too aggressive • Error term does not decrease monotonically GS on Anaheim Network Flow Comparisons for Sioux-Falls Network